Finite Group Databasebeta

← C53×#25521 #27233033

Subgroups — C53×#25521 #27233033

Order 13568.

1118 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3492C211yesmaximal, characteristic, complemented
2C53×#3055C212yesmaximal, complemented
3C53×#2504C211yesmaximal, characteristic, complemented
4C53×#2061C211yesmaximal, characteristic, complemented
5C53×#2040C211yesmaximal, characteristic, complemented
6C53×#2027C211yesmaximal, characteristic, complemented
7C53×#2006C211yesmaximal, characteristic, complemented
8C53×#1883C212yesmaximal, complemented
9C53×#1871C212yesmaximal, complemented
10C53×#1860C212yesmaximal, complemented
11C2×C53×#343C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C53C2212yes—
13C22×M16×C53C2212yes—
14C53×#554—22no—
15C53×#553—22no—
16C53×#552—22no—
17C53×#547—22no—
18C53×#544C2211yescharacteristic
19C53×#539—22no—
20C53×#537C2211yescharacteristic
21C53×#525—22no—
22C53×#525C2212yes—
23C2×#121×C53C2211yescharacteristic
24C53×#520C2212yes—
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#469—22no—
27C53×#468—22no—
28C53×#466—22no—
29C53×#465—22no—
30C53×#428C2212yes—
31C2×#104×C53C2212yes—
32C2×#104×C53C2212yes—
33C53×#418C2212yes—
34C53×#417C2212yes—
35C53×#397—22no—
36C53×#390C2211yescharacteristic
37C53×#390C2211yescharacteristic
38C53×#386—22no—
39C53×#382C2211yescharacteristic
40C53×#382C2211yescharacteristic
41C4×C22⋊C4×C53C2211yescharacteristic
42C53×#343C2211yescharacteristic
43C53×#343C2211yescharacteristic
44C53×#343C2212yes—
45C53×#339C2212yes—
46C53×#339C2212yes—
47C2×#41×C53—22no—
48C2×#41×C53C2312yes—
49C22×D4×C53—21no—
50C22×D4×C53C2311yescharacteristic
51C23×C4×C53D412yes—
52C23×C4×C53C2311yescharacteristic
53AGL(1,ℤ/8)×C53—24no—
54AGL(1,ℤ/8)×C53—42no—
55C2×SD16×C53—22no—
56C2×D8×C53—22no—
57C2×M16×C53—24no—
58C2×M16×C53—22no—
59C2×M16×C53—42no—
60C2×M16×C53C2312yes—
61C2×M16×C53C2312yes—
62C22×C8×C53—22no—
63C4⋊Q8×C53—22no—
64#126×C53—24no—
65#125×C53—24no—
66#125×C53—22no—
67#123×C53—24no—
68#123×C53—24no—
69#123×C53—22no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53—24no—
72C22⋊Q8×C53—22no—
73C22⋊Q8×C53—22no—
74C22⋊Q8×C53D412yes—
75C22⋊Q8×C53D412yes—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53—22no—
81#121×C53D412yes—
82#121×C53D412yes—
83#120×C53—22no—
84C4×Q8×C53—22no—
85C4×D4×C53—24no—
86C4×D4×C53—22no—
87C4×D4×C53—24no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90#117×C53—22no—
91#117×C53—22no—
92#117×C53C2312yes—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53D412yes—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53—22no—
104C2×C42×C53C2311yescharacteristic
105C2×C42×C53C2311yescharacteristic
106#107×C53—22no—
107#106×C53—22no—
108#104×C53—24no—
109#104×C53—42no—
110#103×C53—22no—
111#102×C53—22no—
112#102×C53—22no—
113#102×C53—22no—
114#95×C53—21no—
115#95×C53—22no—
116#95×C53—22no—
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119#41×C53—22no—
120#41×C53—22no—
121#41×C53—22no—
122#41×C53—22no—
123#41×C53—84no—
124C2×Q8×C53—22no—
125C2×Q8×C53C2×D412yes—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—44no—
135C2×D4×C53—41no—
136C2×D4×C53—41no—
137C2×D4×C53C2×D412yes—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—82no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2411yescharacteristic
155SD16×C53—44no—
156D8×C53—44no—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—44no—
161M16×C53#4114yescomplemented
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—44no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D412yes—
176C4⋊C4×C53C2×D412yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—82no—
187C22⋊C4×C53—82no—
188C42×C53—24no—
189C42×C53—22no—
190C42×C53—22no—
191C42×C53—21no—
192C42×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—41no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53#12011yescharacteristic
204C23×C53#12011yescharacteristic
205C23×C53C22×D411yescharacteristic
206Q8×C53—22no—
207Q8×C53—22no—
208Q8×C53—82no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—44no—
212D4×C53—42no—
213D4×C53—42no—
214D4×C53—44no—
215D4×C53—42no—
216D4×C53—42no—
217D4×C53—82no—
218D4×C53—82no—
219D4×C53—82no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—44no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—82no—
245C2×C4×C53—84no—
246C2×C4×C53#12012yes—
247C2×C4×C53C22×D412yes—
248C2×C4×C53C2×#4112yes—
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C8×C53—24no—
252C8×C53—42no—
253#25521C5311yesmaximal, characteristic, direct product factor, Sylow
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—82no—
267C22×C53—82no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53#45412yes—
272C22×C53C2×#12011yescharacteristic
273C22×C53#53211yescharacteristic
274C22×C53#53711yescharacteristic
275C4×C53—22no—
276C4×C53—42no—
277C4×C53—42no—
278C4×C53—42no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#54612yes—
285C4×C53#54712yes—
286#3492C2×C5311yescharacteristic, complemented
287#3055C2×C5312yescomplemented
288#2504C2×C5311yescharacteristic, complemented
289#2061C2×C5311yescharacteristic, complemented
290#2040C2×C5311yescharacteristic, complemented
291#2027C2×C5311yescharacteristic, complemented
292#2006C2×C5311yescharacteristic, complemented
293#1883C2×C5312yescomplemented
294#1871C2×C5312yescomplemented
295#1860C2×C5312yescomplemented
296C2×#343C2×C5311yescharacteristic, complemented
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—22no—
300C2×C53—41no—
301C2×C53—82no—
302C2×C53—161no—
303C2×C53#243011yescharacteristic
304C2×C53C2×#45411yescharacteristic
305C2×C53#301911yescharacteristic
306C2×AGL(1,ℤ/8)C22×C5312yes—
307C22×M16C22×C5312yes—
308#554—22no—
309#553—22no—
310#552—22no—
311#547—22no—
312#544C22×C5311yescharacteristic
313#539—22no—
314#537C22×C5311yescharacteristic
315#525—22no—
316#525C22×C5312yes—
317C2×#121C22×C5311yescharacteristic
318#520C22×C5312yes—
319C22×C4⋊C4C22×C5311yescharacteristic
320#469—22no—
321#468—22no—
322#466—22no—
323#465—22no—
324#428C22×C5312yes—
325C2×#104C22×C5312yes—
326C2×#104C22×C5312yes—
327#418C22×C5312yes—
328#417C22×C5312yes—
329#397—22no—
330#390C22×C5311yescharacteristic
331#390C22×C5311yescharacteristic
332#386—22no—
333#382C22×C5311yescharacteristic
334#382C22×C5311yescharacteristic
335C4×C22⋊C4C22×C5311yescharacteristic
336#343C22×C5311yescharacteristic
337#343C22×C5312yes—
338#343C22×C5311yescharacteristic
339#339C22×C5312yes—
340#339C22×C5312yes—
341C53#2552111yescharacteristic, direct product factor, Sylow
342C2×#41—22no—
343C2×#41C23×C5312yes—
344C22×D4—21no—
345C22×D4C23×C5311yescharacteristic
346C23×C4D4×C5312yes—
347C23×C4C23×C5311yescharacteristic
348AGL(1,ℤ/8)—24no—
349AGL(1,ℤ/8)—42no—
350C2×SD16—22no—
351C2×D8—22no—
352C2×M16—22no—
353C2×M16—24no—
354C2×M16—42no—
355C2×M16C23×C5312yes—
356C2×M16C23×C5312yes—
357C22×C8—22no—
358C4⋊Q8—22no—
359#126—24no—
360#125—24no—
361#125—22no—
362#123—24no—
363#123—24no—
364#123—22no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8D4×C5312yes—
370C22⋊Q8D4×C5312yes—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121D4×C5312yes—
377#121D4×C5312yes—
378#120—22no—
379C4×Q8—22no—
380C4×D4—24no—
381C4×D4—22no—
382C4×D4—24no—
383C4×D4—22no—
384C4×D4—22no—
385#117—22no—
386#117—22no—
387#117C23×C5312yes—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4D4×C5312yes—
392C2×C4⋊C4C23×C5311yescharacteristic
393C2×C4⋊C4C23×C5311yescharacteristic
394C2×C4⋊C4C23×C5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C42C23×C5311yescharacteristic
400C2×C42C23×C5311yescharacteristic
401#107—22no—
402#106—22no—
403#104—24no—
404#104—42no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409#95—21no—
410#95—22no—
411#95—22no—
412C24—21no—
413C24C2×D4×C5311yescharacteristic
414#41—22no—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—84no—
419C2×Q8—22no—
420C2×Q8C2×D4×C5312yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—44no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4C2×D4×C5312yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—82no—
444C22×C4C2×D4×C5311yescharacteristic
445C22×C4C2×D4×C5312yes—
446C22×C4C2×D4×C5312yes—
447C22×C4C2×D4×C5312yes—
448C22×C4C2×D4×C5312yes—
449C22×C4C24×C5311yescharacteristic
450SD16—44no—
451D8—44no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—44no—
456M16#41×C5314yescomplemented
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—44no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4C2×D4×C5312yes—
471C4⋊C4C2×D4×C5312yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23#120×C5311yescharacteristic
499C23#120×C5311yescharacteristic
500C23C22×D4×C5311yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—82no—
504D4—22no—
505D4—22no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—82no—
513D4—82no—
514D4—82no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4#120×C5312yes—
542C2×C4C22×D4×C5312yes—
543C2×C4C2×#41×C5312yes—
544C2×C4H(5,𝔽2)×C5311yescharacteristic
545C2×C42−1+4×C5311yescharacteristic
546C8—24no—
547C8—42no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C53×#45412yes—
566C22C2×#120×C5311yescharacteristic
567C22C53×#53211yescharacteristic
568C22C53×#53711yescharacteristic
569C4—22no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4C53×#54612yes—
579C4C53×#54712yes—
580C2—21no—
581C2—21no—
582C2—22no—
583C2—41no—
584C2—82no—
585C2—161no—
586C2C53×#243011yescharacteristic
587C2C2×C53×#45411yescharacteristic
588C2C53×#301911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C53×#3492C2semidirect
2. C53×#3055C2semidirect
3. C53×#2504C2semidirect
4. C53×#2061C2semidirect
5. C53×#2040C2semidirect
6. C53×#2027C2semidirect
7. C53×#2006C2semidirect
8. C53×#1883C2semidirect
9. C53×#1871C2semidirect
10. C53×#1860C2semidirect
11. C2×C53×#343C2semidirect
161. M16×C53#41semidirect
253. #25521C53direct (N × H)
286. #3492C2×C53semidirect
287. #3055C2×C53semidirect
288. #2504C2×C53semidirect
289. #2061C2×C53semidirect
290. #2040C2×C53semidirect
291. #2027C2×C53semidirect
292. #2006C2×C53semidirect
293. #1883C2×C53semidirect
294. #1871C2×C53semidirect
295. #1860C2×C53semidirect
296. C2×#343C2×C53semidirect
341. C53#25521direct (N × H)
456. M16#41×C53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C53×#34922
C53×#30552
C53×#25042
C53×#20612
C53×#20402
C53×#20272
C53×#20062
C53×#18832
C53×#18712
C53×#18602
C2×C53×#3432
#2552153